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<p><span class="autocomment">growthexperiments-addlink-summary-summary:3|0|0</span></p> <p><b>New page</b></p><div>{{for|the generalization of hypergeometric series |MacRobert E function}}<br /> In [[mathematics]], &#039;&#039;&#039;E-functions&#039;&#039;&#039; are a type of [[power series]] that satisfy particular arithmetic conditions on the coefficients. They are of interest in [[transcendental number theory]], and are closely related to [[G-function (power series)|G-function]]s.<br /> <br /> ==Definition==<br /> <br /> A power series with coefficients in the field of algebraic numbers<br /> <br /> :&lt;math&gt;f(x)=\sum_{n=0}^\infty c_n \frac{x^n}{n!} \in \overline{\mathbb{Q}}[\![x]\!]&lt;/math&gt;<br /> <br /> is called an &#039;&#039;&#039;{{math|1=&#039;&#039;E&#039;&#039;}}-function&#039;&#039;&#039;&lt;ref&gt;Carl Ludwig Siegel, &#039;&#039;Transcendental Numbers&#039;&#039;, p.33, Princeton University Press, 1949.&lt;/ref&gt; if it satisfies the following three conditions:<br /> <br /> * It is a solution of a non-zero [[linear differential equation]] with polynomial coefficients (this implies that all the coefficients {{math|1=&#039;&#039;c&lt;sub&gt;n&lt;/sub&gt;&#039;&#039;}} belong to the same [[algebraic number field]], {{math|1=&#039;&#039;K&#039;&#039;}}, which has [[Degree of a field extension|finite degree]] over the rational numbers);<br /> * For all &lt;math&gt; \varepsilon&gt;0&lt;/math&gt;,&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;math&gt;\overline{\left|c_n\right|}=O\left(n^{n\varepsilon}\right),&lt;/math&gt;<br /> : where the left hand side represents the maximum of the absolute values of all the [[Conjugate element (field theory)|algebraic conjugates]] of {{math|1=&#039;&#039;c&lt;sub&gt;n&lt;/sub&gt;&#039;&#039;}};<br /> * For all &lt;math&gt; \varepsilon&gt;0&lt;/math&gt; there is a sequence of natural numbers {{math|1=&#039;&#039;q&#039;&#039;&lt;sub&gt;0&lt;/sub&gt;, &#039;&#039;q&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;, &#039;&#039;q&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;,...}} such that {{math|1=&#039;&#039;q&lt;sub&gt;n&lt;/sub&gt;c&lt;sub&gt;k&lt;/sub&gt;&#039;&#039;}} is an [[algebraic integer]] in {{math|1=&#039;&#039;K&#039;&#039;}} for {{math|1=&#039;&#039;k&#039;&#039; = 0, 1, 2,..., &#039;&#039;n&#039;&#039;}}, and {{math|1=&#039;&#039;n&#039;&#039; = 0, 1, 2,...}} and for which &lt;math&gt;q_n=O\left(n^{n\varepsilon}\right). &lt;/math&gt;<br /> <br /> The second condition implies that {{math|1=&#039;&#039;f&#039;&#039;}} is an [[entire function]] of {{math|1=&#039;&#039;x&#039;&#039;}}.<br /> <br /> ==Uses==<br /> <br /> {{math|1=&#039;&#039;E&#039;&#039;}}-functions were first studied by [[Carl Ludwig Siegel|Siegel]] in 1929.&lt;ref&gt;C.L. Siegel, &#039;&#039;Über einige Anwendungen diophantischer Approximationen&#039;&#039;, Abh. Preuss. Akad. Wiss. &#039;&#039;&#039;1&#039;&#039;&#039;, 1929.&lt;/ref&gt; He found a method to show that the values taken by certain {{math|1=&#039;&#039;E&#039;&#039;}}-functions were [[algebraically independent]]. This was a result which established the algebraic independence of classes of numbers rather than just [[linear independence]].&lt;ref&gt;Alan Baker, &#039;&#039;Transcendental Number Theory&#039;&#039;, pp.109-112, Cambridge University Press, 1975.&lt;/ref&gt; Since then these functions have proved somewhat useful in [[number theory]] and in particular they have application in [[Transcendental numbers|transcendence]] proofs and [[differential equations]].&lt;ref&gt;[[Serge Lang]], &#039;&#039;Introduction to Transcendental Numbers&#039;&#039;, pp.76-77, Addison-Wesley Publishing Company, 1966.&lt;/ref&gt;<br /> <br /> ==The Siegel–Shidlovsky theorem==<br /> <br /> Perhaps the main result connected to {{math|1=&#039;&#039;E&#039;&#039;}}-functions is the Siegel–Shidlovsky theorem (also known as the Siegel and Shidlovsky theorem), named after [[Carl Ludwig Siegel]] and Andrei Borisovich Shidlovsky.<br /> <br /> Suppose that we are given {{math|1=&#039;&#039;n&#039;&#039;}} {{math|1=&#039;&#039;E&#039;&#039;}}-functions, {{math|1=&#039;&#039;E&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;(&#039;&#039;x&#039;&#039;),...,&#039;&#039;E&#039;&#039;&lt;sub&gt;&#039;&#039;n&#039;&#039;&lt;/sub&gt;(&#039;&#039;x&#039;&#039;)}}, that satisfy a system of homogeneous linear differential equations<br /> :&lt;math&gt;y^\prime_i=\sum_{j=1}^n f_{ij}(x)y_j\quad(1\leq i\leq n)&lt;/math&gt;<br /> where the {{math|1=&#039;&#039;f&lt;sub&gt;ij&lt;/sub&gt;&#039;&#039;}} are [[Rational function|rational functions]] of {{math|1=&#039;&#039;x&#039;&#039;}}, and the coefficients of each {{math|1=&#039;&#039;E&#039;&#039;}} and {{math|1=&#039;&#039;f&#039;&#039;}} are elements of an algebraic number field {{math|1=&#039;&#039;K&#039;&#039;}}. Then the theorem states that if {{math|1=&#039;&#039;E&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;(&#039;&#039;x&#039;&#039;),...,&#039;&#039;E&#039;&#039;&lt;sub&gt;&#039;&#039;n&#039;&#039;&lt;/sub&gt;(&#039;&#039;x&#039;&#039;)}} are algebraically independent over {{math|1=&#039;&#039;K&#039;&#039;(&#039;&#039;x&#039;&#039;)}}, then for any non-zero algebraic number {{math|1=α}} that is not a pole of any of the {{math|1=&#039;&#039;f&lt;sub&gt;ij&lt;/sub&gt;&#039;&#039;}} the numbers {{math|1=&#039;&#039;E&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;(α),...,&#039;&#039;E&#039;&#039;&lt;sub&gt;&#039;&#039;n&#039;&#039;&lt;/sub&gt;(α)}} are algebraically independent.<br /> <br /> ==Examples==<br /> <br /> # Any polynomial with algebraic coefficients is a simple example of an {{math|1=&#039;&#039;E&#039;&#039;}}-function.<br /> # The [[exponential function]] is an {{math|1=&#039;&#039;E&#039;&#039;}}-function, in its case {{math|1=&#039;&#039;c&lt;sub&gt;n&lt;/sub&gt;&#039;&#039;&amp;nbsp;=&amp;nbsp;1}} for all of the {{math|1=&#039;&#039;n&#039;&#039;}}.<br /> # If {{math|1=λ}} is an algebraic number then the [[Bessel function]] {{math|1=&#039;&#039;J&#039;&#039;&lt;sub&gt;λ&lt;/sub&gt;}} is an {{math|1=&#039;&#039;E&#039;&#039;}}-function.<br /> # The sum or product of two {{math|1=&#039;&#039;E&#039;&#039;}}-functions is an {{math|1=&#039;&#039;E&#039;&#039;}}-function. In particular {{math|1=&#039;&#039;E&#039;&#039;}}-functions form a [[Ring (mathematics)|ring]].<br /> # If {{math|1=&#039;&#039;a&#039;&#039;}} is an algebraic number and {{math|1=&#039;&#039;f&#039;&#039;(&#039;&#039;x&#039;&#039;)}} is an {{math|1=&#039;&#039;E&#039;&#039;}}-function then {{math|1=&#039;&#039;f&#039;&#039;(&#039;&#039;ax&#039;&#039;)}} will be an {{math|1=&#039;&#039;E&#039;&#039;}}-function.<br /> # If {{math|1=&#039;&#039;f&#039;&#039;(&#039;&#039;x&#039;&#039;)}} is an {{math|1=&#039;&#039;E&#039;&#039;}}-function then the derivative and integral of {{math|1=&#039;&#039;f&#039;&#039;}} are also {{math|1=&#039;&#039;E&#039;&#039;}}-functions.<br /> <br /> ==References==<br /> <br /> &lt;references/&gt;<br /> * {{mathworld|title=E-Function|urlname=E-Function}}<br /> <br /> [[Category:Transcendental numbers]]<br /> [[Category:Algebraic number theory]]<br /> [[Category:Analytic functions]]<br /> [[Category:Analytic number theory]]</div>

imported>AliceH28